Factor the following expression: $5$ $x^2+$ $14$ $x$ $-24$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(-24)} &=& -120 \\ {a} + {b} &=& & & {14} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-120$ and add them together. Remember, since $-120$ is negative, one of the factors must be negative. The factors that add up to ${14}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${20}$ $ \begin{eqnarray} {ab} &=& ({-6})({20}) &=& -120 \\ {a} + {b} &=& {-6} + {20} &=& 14 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-6}x +{20}x {-24} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-6}x) + ({20}x {-24}) $ Factor out the common factors: $ x(5x - 6) + 4(5x - 6) $ Notice how $(5x - 6)$ has become a common factor. Factor this out to find the answer. $(5x - 6)(x + 4)$